A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubangi Saraf
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry base-code of block-length q to a lifted-code of block-length q m , for arbitrary integer m. The construction generalizes the way degree-d, univariate polynomials evaluated over the q-element field (also known as Reed-Solomon codes) are "lifted" to degree-d, m-variate polynomials (Reed-Muller codes). A number of properties are established: Rate The rate of the degree-lifted code is
more » ... roximately a 1 m! -fraction of the rate of the basecode. Distance The relative distance of the degree-lifted code is at least as large as that of the basecode. This is proved using a generalization of the Schwartz-Zippel Lemma to degree-lifted Algebraic-Geometry codes (Lemma 5.6) . Local correction If the base code is invariant under a group that is "close" to being doublytransitive (in a precise manner defined later , cf. Definition 6.1 ) then the degree-lifted code is locally correctable with query complexity at most q 2 . The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes. Taking a concrete illustrating example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.
doi:10.1145/2488608.2488714 dblp:conf/stoc/Ben-SassonGKKS13 fatcat:hwixwvv265aypdsm6qrmbfwnwq